Group #30755513 — subgroups

Subgroups — #30755513

Order 17152.

402 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863801	C₂	1	1	yes	maximal, characteristic
2	#22561197	C₂	1	2	yes	maximal
3	#22558895	C₂	1	2	yes	maximal
4	#22558498	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524043	C₂²	1	2	yes	—
7	#16863801	—	2	1	no	—
8	#16863801	—	2	1	no	—
9	#16863801	—	2	2	no	—
10	#16863449	C₂²	1	1	yes	characteristic
11	C₂×#14523977	C₄	1	4	yes	—
12	C₄×#11761612	C₂²	1	1	yes	characteristic
13	#16863164	—	2	4	no	—
14	#16863141	—	2	4	no	—
15	#16863118	—	2	1	no	—
16	#16863117	—	2	1	no	—
17	#423×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#334×C₆₇	—	2	2	no	—
20	C₂²×Dic₁₃₄	—	2	2	no	—
21	C₂×#11761612	D₄	1	2	yes	—
22	#14524043	—	2	2	no	—
23	#14524043	—	2	2	no	—
24	#14524043	—	2	2	no	—
25	#14524043	—	2	2	no	—
26	C₂×C₄⋊Dic₆₇	—	2	2	no	—
27	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
28	#14523977	—	2	4	no	—
29	#14523977	—	2	4	no	—
30	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
31	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
32	#14523954	—	2	1	no	—
33	C₄×C₆₇⋊C₈	—	2	1	no	—
34	C₄×C₆₇⋊C₈	—	2	1	no	—
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₆₇	—	2	2	no	—
39	C₄⋊C₈×C₆₇	—	2	2	no	—
40	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₄×C₈×C₆₇	D₄	1	2	yes	—
43	C₂×Dic₁₃₄	—	4	2	no	—
44	C₂×Dic₁₃₄	—	4	2	no	—
45	C₂×Dic₁₃₄	—	8	4	no	—
46	C₂²×Dic₆₇	—	2	4	no	—
47	#11761612	—	2	4	no	—
48	C₂×C₆₇⋊C₈	—	2	2	no	—
49	C₂×C₆₇⋊C₈	—	2	2	no	—
50	C₄⋊Dic₆₇	—	2	4	no	—
51	C₄⋊Dic₆₇	—	2	4	no	—
52	C₄⋊Dic₆₇	—	4	2	no	—
53	C₄⋊Dic₆₇	—	4	2	no	—
54	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
57	M₁₆×C₆₇	—	4	4	no	—
58	C₂×C₈×C₆₇	—	2	4	no	—
59	C₂×C₈×C₆₇	—	4	2	no	—
60	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₆₇	#41	1	1	yes	characteristic
66	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
72	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
73	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#137	1	4	yes	—
76	C₂×Dic₆₇	—	4	4	no	—
77	C₂×Dic₆₇	—	8	4	no	—
78	Dic₁₃₄	—	8	4	no	—
79	Dic₁₃₄	—	8	4	no	—
80	C₈×C₆₇	—	2	2	no	—
81	C₈×C₆₇	—	8	2	no	—
82	C₈×C₆₇	#101	1	4	yes	—
83	C₆₇⋊C₈	—	4	2	no	—
84	C₆₇⋊C₈	—	4	2	no	—
85	C₂²×C₆₇	—	2	1	no	—
86	C₂²×C₆₇	—	2	1	no	—
87	C₂²×C₆₇	#390	1	1	yes	characteristic
88	C₂²×C₆₇	#441	1	2	yes	—
89	C₄×C₆₇	—	2	1	no	—
90	C₄×C₆₇	—	2	1	no	—
91	C₄×C₆₇	C₂×#101	1	2	yes	—
92	C₄×C₆₇	#470	1	2	yes	—
93	C₄×C₆₇	#472	1	2	yes	—
94	C₄×C₆₇	#497	1	2	yes	—
95	Dic₆₇	—	8	4	no	—
96	#9810	—	67	1	no	maximal, Sylow
97	C₂×C₆₇	—	2	2	no	—
98	C₂×C₆₇	#1966	1	1	yes	characteristic
99	C₂×C₆₇	#1971	1	1	yes	characteristic
100	C₂×C₆₇	#1978	1	1	yes	characteristic
101	C₂×#500	—	67	1	no	—
102	#1997	—	67	2	no	—
103	#1566	D₆₇	1	1	yes	characteristic
104	#1549	—	67	2	no	—
105	#1451	—	67	1	no	—
106	C₆₇	#9810	1	1	yes	characteristic, complemented, Sylow
107	C₂×C₄⋊Q₈	—	67	2	no	—
108	#500	—	134	1	no	—
109	#500	—	134	1	no	—
110	#500	—	134	2	no	—
111	#431	—	67	1	no	—
112	C₂×#107	—	67	4	no	—
113	#423	C₂×D₆₇	1	2	yes	—
114	C₄×M₁₆	—	67	1	no	—
115	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
116	#343	—	134	4	no	—
117	#334	—	2	2	no	—
118	#332	—	134	4	no	—
119	#322	—	134	1	no	—
120	#322	—	134	1	no	—
121	C₂²×Q₈	—	134	2	no	—
122	C₂×M₁₆	—	2	2	no	—
123	C₂×M₁₆	—	67	2	no	—
124	C₂²×C₈	C₄×D₆₇	1	2	yes	—
125	C₄⋊Q₈	—	134	2	no	—
126	C₄⋊Q₈	—	134	2	no	—
127	C₄⋊Q₈	—	134	2	no	—
128	C₄⋊Q₈	—	134	2	no	—
129	C₂×C₄⋊C₄	—	67	4	no	—
130	C₂×C₄⋊C₄	—	134	2	no	—
131	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
132	#107	—	134	4	no	—
133	#107	—	134	4	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	67	2	no	—
137	C₂²⋊C₈	—	134	1	no	—
138	#97	—	134	1	no	—
139	C₄×C₈	—	134	1	no	—
140	C₄×C₈	—	134	1	no	—
141	C₄×C₈	D₂₆₈	1	1	yes	characteristic
142	C₄×C₈	D₂₆₈	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
144	C₂×Q₈	—	268	2	no	—
145	C₂×Q₈	—	268	2	no	—
146	C₂×Q₈	—	536	4	no	—
147	C₂²×C₄	—	134	4	no	—
148	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
149	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
150	C₂²×C₄	#11761629	1	1	yes	characteristic
151	M₁₆	—	4	4	no	—
152	M₁₆	—	134	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	134	2	no	—
156	C₂×C₈	—	134	2	no	—
157	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
158	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
159	C₄⋊C₄	—	268	2	no	—
160	C₄⋊C₄	—	268	2	no	—
161	C₄⋊C₄	—	134	4	no	—
162	C₄⋊C₄	—	134	4	no	—
163	C₄²	D₄×D₆₇	1	1	yes	characteristic
164	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
165	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
166	C₄²	#11761629	1	1	yes	characteristic
167	C₂³	#14524005	1	1	yes	characteristic
168	Q₈	—	536	4	no	—
169	Q₈	—	536	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	268	4	no	—
173	C₂×C₄	—	536	4	no	—
174	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
175	C₂×C₄	#14524005	1	1	yes	characteristic
176	C₂×C₄	#14524018	1	1	yes	characteristic
177	C₂×C₄	#14524018	1	1	yes	characteristic
178	C₂×C₄	#14524032	1	1	yes	characteristic
179	C₂×C₄	#14524039	1	1	yes	characteristic
180	C₂×C₄	#14524072	1	4	yes	—
181	C₈	—	2	2	no	—
182	C₈	—	8	2	no	—
183	C₈	—	268	2	no	—
184	C₈	—	268	2	no	—
185	C₈	#14523961	1	4	yes	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	#16863297	1	1	yes	characteristic
189	C₂²	#16863502	1	2	yes	—
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	536	4	no	—
193	C₄	C₂×#14523961	1	2	yes	—
194	C₄	#16863655	1	2	yes	—
195	C₄	#16863667	1	2	yes	—
196	C₄	#16863787	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	#22561052	1	1	yes	characteristic
199	C₂	#22561078	1	1	yes	characteristic
200	C₂	#22561117	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863801

C₂

yes

maximal, characteristic

#22561197

C₂

yes

maximal

#22558895

C₂

yes

maximal

#22558498

C₂

yes

maximal, characteristic

C₆₇×#1566

C₂

yes

maximal, characteristic

C₂×#14524043

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#14523977

C₄

yes

—

C₄×#11761612

C₂²

yes

—

—

—

—

#423×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#334×C₆₇

—

C₂²×Dic₁₃₄

—

C₂×#11761612

D₄

yes

—

—

—

—

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂×C₄

yes

—

#14523977

—

#14523977

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

#41

yes

characteristic

C₂³×C₆₇

C₄×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#101

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#390

yes

characteristic

C₂²×C₆₇

#441

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#101

yes

—

C₄×C₆₇

#470

yes

—

C₄×C₆₇

#472

yes

—

C₄×C₆₇

#497

yes

—

Dic₆₇

—

#9810

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

100

C₂×C₆₇

#1978

yes

characteristic

101

C₂×#500

—

102

#1997

—

103

#1566

D₆₇

yes

characteristic

104

#1549

—

105

#1451

—

106

C₆₇

#9810

yes

characteristic, complemented, Sylow

107

C₂×C₄⋊Q₈

—

108

#500

—

134

—

109

#500

—

134

—

110

#500

—

134

—

111

#431

—

112

C₂×#107

—

113

#423

C₂×D₆₇

yes

—

114

C₄×M₁₆

—

115

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

116

#343

—

134

—

117

#334

—

118

#332

—

134

—

119

#322

—

134

—

120

#322

—

134

—

121

C₂²×Q₈

—

134

—

122

C₂×M₁₆

—

123

C₂×M₁₆

—

124

C₂²×C₈

C₄×D₆₇

yes

—

125

C₄⋊Q₈

—

134

—

126

C₄⋊Q₈

—

134

—

127

C₄⋊Q₈

—

134

—

128

C₄⋊Q₈

—

134

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

134

—

131

C₂×C₄²

C₂²×D₆₇

yes

characteristic

132

#107

—

134

—

133

#107

—

134

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

C₂²⋊C₈

—

134

—

138

#97

—

134

—

139

C₄×C₈

—

134

—

140

C₄×C₈

—

134

—

141

C₄×C₈

D₂₆₈

yes

characteristic

142

C₄×C₈

D₂₆₈

yes

characteristic

143

C₄×C₈

C₂²⋊D₆₇

yes

—

144

C₂×Q₈

—

268

—

145

C₂×Q₈

—

268

—

146

C₂×Q₈

—

536

—

147

C₂²×C₄

—

134

—

148

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

149

C₂²×C₄

D₄×D₆₇

yes

characteristic

150

C₂²×C₄

#11761629

yes

characteristic

151

M₁₆

—

152

M₁₆

—

134

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

134

—

156

C₂×C₈

—

134

—

157

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

158

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

159

C₄⋊C₄

—

268

—

160

C₄⋊C₄

—

268

—

161

C₄⋊C₄

—

134

—

162

C₄⋊C₄

—

134

—

163

C₄²

D₄×D₆₇

yes

characteristic

164

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

165

C₄²

C₂×D₂₆₈

yes

characteristic

166

C₄²

#11761629

yes

characteristic

167

C₂³

#14524005

yes

characteristic

168

Q₈

—

536

—

169

Q₈

—

536

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

268

—

173

C₂×C₄

—

536

—

174

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

175

C₂×C₄

#14524005

yes

characteristic

176

C₂×C₄

#14524018

yes

characteristic

177

C₂×C₄

#14524018

yes

characteristic

178

C₂×C₄

#14524032

yes

characteristic

179

C₂×C₄

#14524039

yes

characteristic

180

C₂×C₄

#14524072

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

268

—

184

C₈

—

268

—

185

C₈

#14523961

yes

—

186

C₂²

—

187

C₂²

—

188

C₂²

#16863297

yes

characteristic

189

C₂²

#16863502

yes

—

190

C₄

—

191

C₄

—

192

C₄

—

536

—

193

C₄

C₂×#14523961

yes

—

194

C₄

#16863655

yes

—

195

C₄

#16863667

yes

—

196

C₄

#16863787

yes

—

197

C₂

—

198

C₂

#22561052

yes

characteristic

199

C₂

#22561078

yes

characteristic

200

C₂

#22561117

yes

characteristic

	subgroup	G/H
center	C₂²	#16863297
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1566	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1566	D₆₇
hypocenter	C₆₇	#9810
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂²

#16863297

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1566

C₂

socle

C₂²×C₆₇

#390

prime radical

C₂³×C₆₇

C₄×D₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
106. C₆₇	#9810	semidirect

H ≅ G/N

type

106. C₆₇

#9810

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16863801	2
#22561197	2
#22558895	2
#22558498	2
C₆₇×#1566	2
#9810	67

subgroup

index

C₆₇×#1566

Subgroups — #30755513

Conjugacy classes of subgroups

Complements

Maximal subgroups